Kevin Connors is arguing with Sir Arthur C. Clarke (the guy who invented geostationary orbit, and popularized the concept of space elevators) about the technical viability of the latter. All I can say is that he's a braver man than I. He's also a little confused about orbital mechanics:

...orbit in the Clarke Belt is achieved because the centrifugal force of the orbiting satellite exactly matches the force imparted upon it by gravity.

Well, this is sort of correct, but oversimplifed. In reality, there's no such thing as a centrifugal force, but one can pretend there is in the rotating (non-inertial) reference frame. It's more correct to say that the centripetal acceleration exactly matches that of gravity at that altitude.

Propelling a payload up a tether attached to that satellite would upset that equilibrium. Further, their is the distributed mass of the tether itself to consider. It is therefore necessary that the satellite be in a far lower orbit, in order to maintain tension on the tether.

This is where he goes off the tracks. I don't know why he thinks a lower orbit would be required (or what he means when he says "satellite").

A space elevator is designed to have its center of mass at a point beyond geostationary orbit. The idea is to have a balance between the forces that would provide sufficient tension in the cable. During construction, the anchor would initially be in GEO, but as the cable is dropped from it, it will move upward to keep the CM at GEO altitude, to maintain a geostationary period. Once the cable has reached down to earth, the other end is anchored. At that point, you'd continue to reel it out, but moving the anchor up to increase tension in the cable to whatever was desired, at which point the geostationary orbital period is maintained by being attached to the planet. The old conventional wisdom (if such a phrase makes sense in the context of a concept like this) was that one might use a small asteroid for the anchor. Newer concepts don't require as much mass, but in either case, there will be sufficient mass, at a sufficient supergeostationary altitude to allow motion up and down without major issues.

Indeed, the path the transport vehicle takes to reach the satellite will not be a straight path, as is popularly envisioned, but a great parabolic arc.

Again, I don't know what he means by this, but (also again) the path will depend on the reference frame. From the reference frame of a rotating earth, the path will follow the cable, which is to say straight up to GEO (where the weightless docking station would be, though the elevator structure would continue on to higher altitude, as described above). From an inertial frame, the path would appear to be a spiral, as the car orbits the earth once per day with increasing altitude. There will be some coriolis force on the moving car as a function of its velocity and altitude (as there is in an earthly elevator car), but the tension of the cable will be designed to be sufficient to prevent it from bending it much.

From a basic physics standpoint, the concept is fine, and can be easily simulated, honest.

Posted by Rand Simberg at September 25, 2005 02:21 PM

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Going up!Excerpt: When I read Sir Arthur C. Clarke's The Fountains of Paradise about 20 years ago I considered it a masterful piece of Science Fiction that, unfortunately, told a tale that could never come true. A lot can change in 20Weblog: Small Town VeteranTracked: September 26, 2005 12:30 AM

Going up!Excerpt: When I read Sir Arthur C. Clarke's The Fountains of Paradise about 20 years ago I considered it a masterful piece of Science Fiction that, unfortunately, told a tale that could never come true. A lot can change in 20Weblog: Small Town VeteranTracked: September 26, 2005 12:34 AM

Space ElevatorsExcerpt: Rand Simberg has a good post on the subject, titled Trouble with the Concept, in which he refutes a post by Kevin Connors questioning the validity of the orbital mechanics analysis for the space elevator concept. There is some good discussion in th...Weblog: SpacecraftTracked: September 27, 2005 08:06 AM

Comments

> "orbit in the Clarke Belt is achieved because the centrifugal force of the orbiting satellite exactly matches the force imparted upon it by gravity."

And every other orbit. If he thinks the forces don't match in, say, the moon's orbit, why does the moon stay in its orbit?

The whole idea of the space elevator is that the "centrifugal force" of the cable system exceeds the gravitational force on it. The excess upward force is countered by the earthside anchor and the elevator car. The car will not "disturb the equilibrium," it will just reduce the force that the anchor has to provide.

Now, in order to get centrifugal force greater than the downward force of gravity, you need mass traveling faster than orbital velocity. Orbital velocity is, by definition, the velocity at which they match, after all. Orbital velocity decreases with altitude, since gravity decreases with altitude. Therefore you need mass at altitudes beyond Clarke orbit, where orbital velocity is less, exactly the opposite of what Connors says.

And every other orbit. If he thinks the forces don't match in, say, the moon's orbit, why does the moon stay in its orbit?

Yeah, I didn't catch that. But it's not really every other orbit. It's any circular orbit...

Posted by Rand Simberg at September 25, 2005 03:28 PM

Thank heavens C60 nanotubes have the tensile strength to weight ratio to deal with the forces involved or we'd be pretty screwed, since there is nothing else even on the horizon that we could use for the cable.

C60 is not a nanotube. C60 is a Buckyball (fullerene sphere). But yes, it is a good thing that nanotubes have arrived on the scene, as nothing else has the tensile strength to support a space elevator.

The discussion about the shape of the elevator has nothing to do with the frame of refence, because it is not moving at relativistic speed. Either straight or parabola, it's going to be the same regardless. Reference plane is only important when we talk about forces. So Rand outsmarted himself here a little.

But anyway, it is not so obvious that the elevator ribbon is going to be straight. What shape is it going to be? The answer has to be gotten with something known as "variational calculus", I suspect. Beats me. I got C in that, 25 years ago. It may end straight.

Actually, if we start with something straight, it seems like a solution. There aren't any forces acting on any given piece of the ribbon which aren't along the straight line to the Earth's center (in first approximation). But is this configuration stable? Once the ribbon is bent a little (by Moon's gravity, for example), the tension gets off-center with the Earth gravity. Well, I hope someone solved that equation, at least numerically...

Back to the article, as far as I know, the great parabola is what appears if you put a drop of ink on a rotating paper disk in a middle school class. But I am sure this is NOT how the elevator is going to look, simply because the force on any segment of the ribbon is the sum of gravity and the tension, while the force on the ink drop is the friction against the disk (in inertial plane), which acts in different direction!

Since the tether must accelerate the payload horizontally, there are some lateral forces (i.e. tangent to the earth surface at the tether attachment point).

Initially the payload stands on the ground and goes the circumference of the earth in one day, i.e.: 40 000 km / 24 h. In the end it is in geostationary orbit, travelling 270 000 km / 24 h. Angular velocity stays the same (2pi / 24h) but the radius and thus the path/track/circumference velocity (whatever it's in english) increases almost 7-fold.

You get the similar effect as coriolis force (that causes weather patterns on earth). The path velocity of earth's surface is higher at equator than at higher latitudes, and is zero at the poles. Air near equator has relatively little speed compared to ground but if it moves up north, it turns right since the path velocity tends to stay the same, but the radius from the axis of earth's rotation decreases.

On a space tether, the tether is mostly vertical, but since it is horizontally tensioned by the upward moving payload that must be accelerated, it will probably be slightly > -shaped (when viewed from north) where the payload is at the bottom of the horizontal "V".
The energy increase for the payload comes from the rotational energy of the earth that lessens a bit with every payload sent into space.

A point which I assume most readers of this site understand, but which I haven't seen anyone point out yet, is that the cable will need to be lifted to, or created in, orbit and lowered to the ground, not lifted into place with one end anchored groundside. I've explained my reasoning in more detail at
http://smalltownveteran.typepad.com/posts/2005/09/going_up.html

Mr. Connors may well join that large group of naysayers and luddites that litter history for having digested their own feet.

People in the late 1800s KNEW man would never fly.

People in the 1950s and 1960s KNEW we would never get into space. Some still refuse to believe we went to the moon

Mr. Connors and others KNOW space platforms will never work.

I expect money will be the issue here. Maybe this will be the time when private venture pushes "gubments" out of the space business. All we need is Queen Isabellas crown to pay for the damn thing and it will get built.

Slashdot this morning had a piece on a "first step" experiment. A fiber glass ribbon was hoisted by a balloon to 1000 feet. A robot climbed the ribbon successfully. They plan to go for a 1 mile climb (5000 feet). Granted that the process needs to be scaled up by a factor of about 100000, its still a first step.
Question. What does the real climber use for power? Can batteries have enough juice to hoist their own weight 22000 miles? Let alone payload. Can electricity be fed up the tether to power the climber? Will said electricity ionize the low pressure atmosphere at say 200 000 feet and arc over? Do sliding electrical pickups work in vacuum? Will Can a climber carry enough oxygen to run a combustion engine to power itself? Are fuel cells practical?

To maximize cargo, they're looking at beamed power from the platform. The cars would have a large solar panel facing down.

They're looking at a nanometer-class laser of some sort that I read doesn't exist yet for optimum efficiency. I'd recommend that they put some less-efficient designs on paper with existing lasers if they haven't already.

Posted by Big D at September 26, 2005 07:54 AM

I'm still not convinced that this will be a viable technology. My concerns are mainly financial - will you be able to lift enough mass into orbit (limitted by the flight rate and the maintainence) at a cost enough lower than a (also nanotube-utilizing) rocket could achieve, such that the mass times the decrease in cost is less than the difference in price between the rocket and the elevator.

What is an achievable flight rate? Call it 2 flights per week (400 km/hr on a hoist). What is the mainaintence load? Well, this seems to lead to a new cable required every week - but since it hasn't been done, let's pretend that it never fails, so lets say 20 years (basically anything we do now is financially insignificant in 20 years). So that is ~2100 flights maximum. Just for simplicity, assume a 10 ton to orbit load. Assume also a $200K per ton flight cost, not including amortization. That means the total flight cost over 20 years is $420M.

Pretend rockets do not change drastically (nanotube propellant tanks don't work for some reason) - rocket costs are $10M per ton in orbit. So flying rockets to do the job will cost $21B, right? So any elevator that costs less than $20B should be built, right?

Unfortunately, wrong. The rocket's cost is spread out over 20 years - and since a dollar today is worth more than a dollar tommorow (otherwise lo_ans would not exist), the rocket's cost is actually far lower. Basically, take $1B per year over 20 years and find the net present value (NPV in Excel), using your favorite discount rate (startups and risky ventures use ~20%, banks use ~10%). At 20%, your number is $4.87B. Even worse, the income your elevator makes is also spread out over 20 years (so it almost disappears as well), while the incured cost is all up front (so it is counted fully). Can one build an elevator for $4.5 B? Compare what you think about that question's answer to this: What would rockets really cost if they could use nanotubes, and we could spend $4B in R&D on them?

The cost of capital is what I believe kills the space elevator. It also is why very few people invest in billion dollar rocket research - the status quo tends to beat that. The numbers really lead to a requirement for incremental development, which doesn't really exist for a space elevator.

OTOH, I'm sure NASA would love to build one. I think government agencies have an inverse cost of capital - the more money you spend now, the more money you have later!

A bit red-faced after that sign-inversion. But those sort of things happen when you're bouncing these things around in your head.

In any event, I never said the space elevator wouldn't work, only that it wouldn't be quite as currently envisioned. I still feel that the cable will arc into space, And it appears this effect has already been contemplated for the mass of the transport vehicle ("climber", if you will), but not for the distributed mass of the cable itself.

Kevin, you can "feel" whatever you want, but people who have actually done the analysis know that it will be a straight line, absent any disturbances from moving cars. If you are going to argue with Arthur Clarke and others who do this for a living, and want to be taken seriously, you're going to actually have to do the math, and show your work.

Posted by Rand Simberg at September 26, 2005 11:33 AM

Despite my own usage of the word, feel is not so good a word as suspect, Rand. But point taken, nonetheless.

However, your implication that mathematical integrity implies fact is totally erroneous. As, if the underlying model is flawed, the analysis will be flawed, despite the fact that the math "works".

Anyway, on further investigation, I believe that, even if correct, my assertion about the tether arcing, by virtue of it's own mass, is trivial, as the differential mass between the counterweight and the tether is so great.

However, I ask, what is the principle inertial sink? Contemporary thinking seems to be that it is the counterweight. In such case (as the counterweight is of finite mass), the counterweight will suffer orbital decay with continued use, and require occasional reboost. In the interim, there will be arcing in the tether.

OR, is the inertial sink the Earth itself? As seems to be asserted in this paper by J. Pearson, and seems more logical to me, as the mass of the Earth is virtually infinite in this context.

But, if that is the case, it would imply energy transfer up the tether to the counterweight. And that would entail arcing of the tether.

Again, that arcing may be trivial. But I have yet to see where it is addressed at all.

However, your implication that mathematical integrity implies fact is totally erroneous. As, if the underlying model is flawed, the analysis will be flawed, despite the fact that the math "works".

I've no idea what you think you mean by the "underlying model." The underlying model is Newtonian physics. There is no reason to think that relativity, either general or special, need be invoked. And under that model, the math works fine, so I don't know what it is you're challenging here. What is it that you think "flawed" by Newtonian physics, that justifies your objections?

I also don't know what you mean by "inertial sink."

And why would adding energy to a tether intrinsically result in arcing?

People who study these things have a language in which to discuss them, with precise definitions for words, and mathematical equations. So far, you seem to be speaking a different language, your feelings and suspicions aside.

Posted by Rand Simberg at September 26, 2005 05:12 PM

I think by "inertial sink", he means which part of the structure "pays" the energy bill for accellerating the car as it goes up. The answer, as you already stated, is the planet (Edwards' original FAQ joked that this would add an hour to the day if we shipped Australia to space).

The only reasonable technical question that I've heard of is SDB's concern that the transit-induced oscillations would be an issue. It sounds in the more recent FAQs like that is being addressed with a combination of platform and anchor movements to dampen transit oscillations. This means that fuel must be delivered to the counterweight, but even so, I don't see that as a significant (13 tons of fuel, with ion drives, gets you...) cost issue.

As for the financial side of it... that's a harder question, but don't forget that the second elevator can be made even cheaper, by putting the initial spool up with the original elevator. Also, your insurance costs should go way down after the strength of the elevator is proven, because the payload isn't sitting on top of something that can go boom (by accident or by range control if it goes off course).

Posted by Big D at September 26, 2005 10:50 PM

In the 1950s, the folks at JPL who were developing the first US satellite, Explorer I, did all the math. At that time, everyone knew that a rigid body is stable when spinning about either its minor (like a football) or major (like a frisbee) axis. Everyone also knew that this stability was marginal, since the eigenvalues are on the imaginary axis (Lyapunov hadn't really reached our shores yet). And everyone knew that adding a little damping pushes those eigenvalues off the imaginary axis. And everyone knew that that push would put them in the left half plane and presto! asymptotic stabilty.

So the designers designed Explorer I to spin about its minor axis, and added some whip antennas to provide energy dissipation to achieve asymptotic stability.

As described by Peter Likins, "Professor Ronald Bracewell, a radio astronomer at Stanford, had deduced from Sputnik's signals that it was in a stable spin about an axis of symmetry and, from his understanding of arguments derived initially for the rotational dynamics of galaxies, he concluded that Sputnik must be spinning about an axis of maximum moment of inertia." Bracewell tried to contact JPL, but secrecy prevailed, and his deductions were not incorporated into the Explorer I design.

Within 90 minutes or so, Explorer I's minor-axis spin became a major-axis spin, and it tumbled about an axis perpendicular to the intended axis.

That's just one example of a spacecraft whose dynamic behavior was different from the designers' intentions. Many space tether missions also fit into that category.

I'm optimistic about the potential for space elevators. I've done the first-order analysis myself, and nothing I've seen published seems to be obviously incorrect, but I expect I'm going to remain an optimistic skeptic a little while longer.

The car makes a dent in the cable since it must be ACCELERATED. The cable does not accelerate horizontally, thus it doesn't bend or dent or do anything if compared to the car. I thought I was clear on that one.

If the additional horizontal velocity of the car comes from just Earth pulling the car to the side, there is no need for arcing, although there are details[1]: instead the car just trails a little behind the point directly above the equatorial land anchor. Actually, the space anchor originally trails too since the natural orbital angular velocity at over GEO is slower than 2pi/24h so it must be dragged on by the Earth anchor, but after it gets the fast path velocity (dragged by Earth's rotation) it can straighten up to be on the same line as earth's center and the land anchor. Now only the cars sent up still need to be accelerated so they are pulled up to speed by the slightly skewed cable. They may move the space anchor somewhat to the west (and land anchor if it's at sea), and it can probably be tuned so that the space anchor is a bit behind all the time by sendind cars up, but kept in orbit because of the pull of horizontal acceleration from earth.

[1] When the tether is moving up or down, then it'll arc because of the coriolis effect: if the path speed (energy) stays the same (it does if there are no external forces), but the radius is increased/decreased, then the angular velocity decreases/increases. So if you lower an untensioned cable from GEO (or above) it will get curved but that's because of the energy it has, not because of any horizontal forces and it can be straightened and stay straight.

"The only reasonable technical question that I've heard of is SDB's concern that the transit-induced oscillations would be an issue."

Indeed. I realized that's what I was getting at late last night, when I thought of the system in a dynamic, rather than static manner. It's nice to know that the matter is being addressed (and, wouldn't ya' know it - in fundamentally the way I prescribed).

Something else that occurred to me: It would seem that, with the massive tension the tether is subjected to, the energy released from even a single filament snapping could cause a cascade failure, taking the rest of the tether with it.

...the energy released from even a single filament snapping could cause a cascade failure, taking the rest of the tether with it.
If there only a few filaments, yes. But just like a terrestrial (as opposed to a transterrestrial :^) elevator cable, there will be enough redundant strands so that the fracture of a reasonably small number will have no noticeable effect.

You don't understand what I mean by cascade, chris. A stressed mechanical member posesses stored energy, like a spring. When a tensioned member snaps, that energy is released - quite suddenly. With this amount of tension - I'm reading numbers like 50x that of an equivalent steel cable - so much energy would be released by a single filament snapping as to damage all adjacent filaments. They would then snap, damaging even more filaments...

There's supposed to be some sort of cross-bracing along the length of the ribbon to mitigate that and preserve structural integrity of the rest of the ribbon if a hole gets poked in it.

It's not a bunch of individual strands of nanotubes so much as it is a fabric of composites that consist mostly of nanotubes.

Posted by Big D at September 27, 2005 09:51 PM

Sorry, but the ribbon can not be straight. Basic physics says it must be a bow. Any perpendicular force on a straight line under tension is infinite. This is how you pull a car out of a ditch when all you've got is a rope and a stout tree. Actually, SDB did an article covering some of the problems with an elevator but never got to the second part he promised (which I've been hoping would happen one day, but expect it will never.)

I also wonder if any change in spin could occur? Perhaps we could adjust for that leap second? ;) It's a silly question but has anyone done the math?

All that aside, I would cheer any company that overcomes the difficulties.

I have similar concerns as those shared by Kevin L. Connors. I think it is equivalent but let me voice it in the following manner.
As the climber is sent up, it steals angular momentum from the entire space elevator (tether, anchor and all). This slows down the orbit or the entire structure. Since it is no longer orbiting at GEO, the tether will begin to wrap around the earth, getting shorter and shorter, inventually bringing down the whole thing.
Now this scenario would not happen if the angular momentum was taken from the earth instead of the elevator. The only way i can think of to guarantee this is if the tether was completely rigid, and somehow forced to be at 90 degrees to the ground where it is anchored.
However since this is impossible, why would the climber prefer to take angular momentum from the earth instead of the elevator? What mechanism guarantees this? I would like to see this issue addressed. I am still reading Edwards book. Does anyone know where and if he talks about this?